#### Question

If the vectors \[\hat{i} - 2x \hat{j} + 3y \hat{k} \text{ and } \hat{i} + 2x \hat{j} - 3y \hat{k}\] are perpendicular, then the locus of (*x*, *y*) is

(a) a circle

(b) an ellipse

(c) a hyperbola

(d) None of these

#### Solution

(b) an ellipse

\[\text{ Let }, \vec{a} = \hat{ i } - 2x \hat{j} + 3y \hat{k} \text{ and } \vec{b} = \hat{i} + 2x \hat{j} - 3y \hat{k} \]

\[\text{ It is given that the vectors are perpendicular. So, their dot product is zero }.\]

\[ \vec{a} . \vec{b} = 0\]

\[ \Rightarrow \left( \hat{i} - 2x \hat{j} + 3y \hat{k} \right) . \left( \hat{i} + 2x \hat{j} - 3y \hat{k} \right) = 0\]

\[ \Rightarrow 1 - 4 x^2 - 9 y^2 = 0\]

\[ \Rightarrow 4 x^2 + 9 y^2 = 1\]

\[\text{ Dividing both sides by } 36, \text{ we get }\]

\[\frac{x^2}{9} + \frac{y^2}{4} = 1\]

\[\text{ This is an ellipse }.\]